Skip to content
ElecDatum

AC impedance calculation

Impedance is a fundamental magnitude in the analysis of alternating current (AC) circuits that generalizes the concept of resistance. It quantifies the total opposition presented by a circuit to the flow of a sinusoidal current, combining the effects of ohmic resistance and reactance (inductive and capacitive). It is expressed as a complex number whose unit in the International System is the ohm (Ω).

Definition and complex representation of impedance

Section titled “Definition and complex representation of impedance”

In an AC circuit in permanent sinusoidal steady state, the impedance (Z) is defined as the ratio between the voltage phasor (V) and the current phasor (I). Its typical value in power electronics applications can range from a few milliohms to several kiloohms. The most widely used complex representation is the Cartesian form:

Z = R + jX

Symbol Quantity Unit
Z Complex impedance Ω
R Resistance (real part) Ω
X Reactance (imaginary part) Ω
j Imaginary unit ((\sqrt{-1})) –

The real part (R) corresponds to the electrical resistance of the circuit, while the imaginary part (X) is the reactance, which can be inductive ((X_L) positive) or capacitive ((X_C) negative).

Components of impedance: resistance and reactance

Section titled “Components of impedance: resistance and reactance”

In an AC circuit, the resistive component dissipates energy in the form of heat and is independent of frequency, while the reactive component stores and returns energy without dissipating it and its value varies with frequency. Resistance is measured in ohms (Ω), as is reactance.

Resistance is the opposition to the movement of charges due to collisions in the material structure; for a metallic conductor at 20 °C / 68 °F, a typical resistance can be 10 Ω. Reactance, on the other hand, arises from the magnetic (inductance) and electric (capacitance) fields generated by the circuit elements. While in direct current reactance has no effect, in alternating current it introduces a phase shift between voltage and current.

Inductive and capacitive reactance

Section titled “Inductive and capacitive reactance”

The inductive reactance of a 100 mH coil at 50 Hz is approximately 31.4 Ω, while the capacitive reactance of a 100 µF capacitor at the same frequency is 31.8 Ω. Both depend directly on the angular frequency (\omega = 2\pi f) and on the values of inductance (L) and capacitance (C).

Inductive reactance:

X_L = 2\pi f L = \omega L

Capacitive reactance (magnitude):

X_C = \frac{1}{2\pi f C} = \frac{1}{\omega C}

In complex notation, the impedance of a pure inductor is (Z_L = j\omega L) and that of a pure capacitor is (Z_C = -j\frac{1}{\omega C} = \frac{1}{j\omega C}). By convention, inductive reactance is considered positive and capacitive reactance negative, which determines the sign of the phase angle.

Parameter Symbol Unit Relationship with frequency
Inductance L Henry (H) Reactance directly proportional to (f)
Capacitance C Farad (F) Reactance inversely proportional to (f)
Angular frequency (\omega) rad/s (\omega = 2\pi f)

Magnitude and phase angle of impedance

Section titled “Magnitude and phase angle of impedance”

For an impedance with (R = 10\ \Omega) and (X = 5\ \Omega), its magnitude is 11.18 Ω and the phase angle is 26.6°. The magnitude (or modulus) of impedance is calculated by:

|Z| = \sqrt{R^{2} + X^{2}}

The phase angle (\theta) represents the phase shift between voltage and current:

\theta = \arctan\left(\frac{X}{R}\right)

Symbol Meaning Unit / Range
( |Z| ) Magnitude of impedance Ω
(\theta) Phase angle Degrees (°) or radians
(R) Resistance Ω
(X) Net reactance ((X_L - X_C)) Ω

When the net reactance is positive (predominantly inductive), (\theta > 0) and the current lags the voltage. If the reactance is negative (predominantly capacitive), (\theta < 0) and the current leads the voltage.

Calculation of equivalent impedances in series and parallel

Section titled “Calculation of equivalent impedances in series and parallel”

The total impedance of two elements in series (Z_1 = 3 + j4\ \Omega) and (Z_2 = 1 - j2\ \Omega) is (4 + j2\ \Omega), with a magnitude of 4.47 Ω. The rules for combining impedances are analogous to those for resistors, but using complex algebra.

Series connection:

Z_{eq} = Z_{1} + Z_{2} + \dots + Z_{n}

Parallel connection:

\frac{1}{Z_{eq}} = \frac{1}{Z_{1}} + \frac{1}{Z_{2}} + \dots + \frac{1}{Z_{n}}

For the particular case of two impedances in parallel, (Z_{eq} = \frac{Z_1 Z_2}{Z_1 + Z_2}).

In series RLC circuits, the equivalent impedance is obtained by summing the contributions:

Z_{series} = R + j\left(\omega L - \frac{1}{\omega C}\right)

In parallel RLC circuits, admittances ((Y = 1/Z)) are used:

Y_{parallel} = \frac{1}{R} + j\left(\omega C - \frac{1}{\omega L}\right)

Example of impedance calculation in a series RL circuit

Section titled “Example of impedance calculation in a series RL circuit”

A series RL circuit is available with a resistance of 40 Ω and a coil of 0.1 H connected to a 60 Hz source. The inductive reactance is calculated as (X_L = 2\pi \cdot 60 \cdot 0.1 \approx 37.7\ \Omega). The total impedance in complex form is:

Z = 40 + j37.7\ \Omega

The magnitude of the impedance is ( |Z| = \sqrt{40^2 + 37.7^2} \approx 54.5\ \Omega) and the phase angle (\theta = \arctan(37.7 / 40) \approx 43.3^\circ). This positive angle indicates that the current lags the voltage, characteristic behavior of an inductive circuit. If the supply voltage had an RMS value of 120 V, the RMS current would be (I = 120\ \text{V} / 54.5\ \Omega \approx 2.2\ \text{A}).

Frequently Asked Questions (FAQ)

Section titled “Frequently Asked Questions (FAQ)”

What is the difference between resistance and impedance?

Section titled “What is the difference between resistance and impedance?”

Resistance is the opposition to current flow in direct current and depends only on the material and geometry; impedance is the opposition in alternating current, including reactive effects that depend on frequency and introduce a phase shift between voltage and current.

How is inductive reactance calculated?

Section titled “How is inductive reactance calculated?”

Inductive reactance is calculated with the formula (X_L = 2\pi f L), where (f) is the frequency in hertz and (L) the inductance in henrys.

What does a positive or negative phase angle mean in impedance?

Section titled “What does a positive or negative phase angle mean in impedance?”

A positive phase angle ((X > 0)) indicates a circuit with inductive behavior, where the current lags the voltage; a negative angle ((X < 0)) indicates a capacitive circuit, where the current leads the voltage.

How are impedances added in series and parallel?

Section titled “How are impedances added in series and parallel?”

In series they are added directly as complex numbers ((Z_{eq} = Z_1 + Z_2 + \dots)). In parallel, their admittances (inverses) are added, or for two impedances, use (Z_{eq} = (Z_1 Z_2)/(Z_1 + Z_2)).

Why does impedance vary with frequency?

Section titled “Why does impedance vary with frequency?”

Because inductive and capacitive reactance depend on the angular frequency (\omega = 2\pi f). As frequency increases, (X_L) increases and (X_C) decreases, modifying the net reactance and therefore the total impedance.

What instrument is used to measure impedance?

Section titled “What instrument is used to measure impedance?”

An impedance analyzer or an LCR bridge is used, which applies a test signal at different frequencies and measures the response in magnitude and phase.

References

Section titled “References”